Tensor network approximation of Koopman operators

The paper’s Proposition 9 rigorously proves that, under assumptions (K1)–(K7) and (V1)–(V6), the Fock-space approximation f^{(t)}_{τ,n} defined via Af,τ,n = Δ_{n-1}Mf,τΔ^*_{n-1} and evolved by the lifted dynamics converges to the classical expectation Ep(U^t f) as τ→0+, with a short proof based on multiplicativity of K^*_τ and strong convergence K^*_τU^{t*}_τ→U^tι on H1 (Lemma 6) . By contrast, the candidate solution incorrectly collapses the Fock-space expectation to a one-particle expectation on H_τ that is independent of n for all t, contradicting the paper’s own derivation, which shows n-dependence for t≠0 and notes independence only at t=0 . The candidate also uses an unsubstantiated factorization K_τ = T_τK_{τ/2} and U^t_τ = T_τe^{tV_τ}T_τ^*, which are not part of the paper’s framework; the paper instead uses the identity K_{τ/2}e^{tV_τ}K_{τ/2} = K^*_τe^{tW_τ}K_τ and the strong dynamical limit to U^t on H . Therefore, while the conclusion matches the paper, the model’s reasoning hinges on false equalities and missing hypotheses.